Observe that in the special case of the join construction, we can take \(t\)  to be the inclusion map
\([0,1]\rightarrow \mathbb {C}\) . Moreover, as explained in the next section, the equivariant join
comodule algebra \(P\circledast ^\delta H\) 
becomes piecewise trivial, and Theorem REF  follows from [1]}.
However, even in this special case of the equivariant join comodule algebra,
our non-unital strong-connection formula is differently constructed than the complicated strong-connection
formula in [1]}. This might be very important for index pairing computations involving
concrete strong-connection formulas.
Finally, let us remark that a fully fledged \(C^*\) -algebraic approach to Theorem REF 
can be found in [3]}.
